CultLib/packages/cultmath/docs/design.md

Design

CultMath exists to make renderer math readable in every place it needs to live: Rust hot kernels, C# host code, and HLSL shader code.

Parity Target: HLSL

HLSL is the semantic authority for every type and intrinsic HLSL defines. The goal is source compatibility: HLSL-shaped math compiles as C# under using static CultMath.math;. C# forces float literal suffixes (0.5f); the rest of the spelling should survive the move.

That makes these HLSL rules, not options:

• Every swizzle exists. float2/3/4, int2/3/4, and bool2/3/4 expose every read swizzle of length 2 to 4, repeats included (v.wwxy), in both xyzw and rgba naming (never mixed), plus single-component r/g/b/a. double2/3 get lengths 2 and 3 only, because there is no double4. Components and swizzles are writable: v.xz = f2;, c.rgb *= 0.5f;, and v[1] = s; work. Setters exist only for non-repeating masks, because HLSL rejects v.xx = ....
• Every HLSL mixed constructor exists, as a struct constructor and as a math function: float4(float2, float2), float4(float, float2, float), float3(float, float2), the copy form float3(float3), and the rest, for the float, int, and bool families (double gets constructors only; a math.double3 function would hide the double3 type name).
• Swizzles and mixed constructors are generated. tools/generate-swizzles.ps1 writes the checked-in src/CultMath/Swizzles.g.cs (partial structs plus a partial math). Edit the generator, rerun it, and commit both; never edit the output by hand.
• Comparison operators (<, >, <=, >=, ==, !=) are component-wise and return bool2/bool3/bool4. Reduce them with any and all. Equals/GetHashCode stay value-based so vectors still work as dictionary keys.
• select(condition, whenTrue, whenFalse) uses HLSL's argument order.
• Matrices are row-major: constructors take rows, m[i] returns a reference to row i (so m[1][2] = s; writes one element), _mRC is row R column C, and products go through mul (mul(m, v) treats v as a column, mul(v, m) as a row). * on matrices would be component-wise in HLSL, so CultMath does not define it.
• m[r][c] = v writes through locals, array elements, fields of classes, and ref locals. Through a readonly field, an in parameter, a property (auto-properties included), or a static readonly value such as float3x3.identity, C# takes a defensive copy: the write compiles and is silently lost. Copy the matrix to a local, write to it, and assign it back.
• Scalar/vector arithmetic and comparisons have explicit overloads on both sides rather than leaning on implicit scalar splat.
• Intrinsics return HLSL's types and edge cases: sign returns int/intN (NaN gives 0); step(y, x) is x < y ? 0 : 1 (NaN gives 1), which is how dxc lowers it, in C# and in the HLSL mirror; float and double min and max, and float clamp and saturate, follow the DXIL FMin/FMax/Saturate operations dxc emits for them: min(a, b) is a < b ? a : b, max(a, b) is a >= b ? a : b, a NaN operand returns the other (so saturate(NaN) is 0), clamp(x, a, b) is min(max(x, a), b), and saturate(x) is min(1, max(0, x)), which maps -0 to +0; any/all accept numeric vectors (a component is true when it is not 0); min, max, and abs on int vectors return int vectors; int clamp is IMin(IMax(x, a), b), so inverted bounds return b.
• Same-size vector conversions (intN(floatN), floatN(intN), boolN(floatN), boolN(intN), floatN(boolN), intN(boolN)) exist as constructors and math functions and follow dxc: float to int is fptosi, truncating toward zero; numeric to bool is != 0 (fcmp une, so NaN is true); bool to numeric is 0 or 1. DXIL leaves float-to-int undefined for NaN and out-of-range values, and so does C#; CultMath does not pin them.
• normalize(x) is x * Rsqrt(Dot(x, x)), which is how dxc lowers it, and DXIL.rst defines Rsqrt as 1 / sqrt(src). A zero vector therefore normalizes to NaN (0 * inf), and a NaN component makes every component NaN. (The special-value table under Rsqrt in DXIL.rst is a copy of the Round table and contradicts that definition, so CultMath follows the definition.) length and distance are Sqrt of the summed squares, so a zero vector gives 0. Callers that can meet zero-length vectors guard them before calling normalize.
• dxc demotes double frac, exp, lerp, and floor to float. CultMath's double overloads keep double precision instead: frac(double) is x - floor(x) in double. That is a deliberate divergence for CPU simulation time, which shaders do not carry.

dxc marks float arithmetic and compares fast (no NaNs) unless a value is precise, so drivers may optimize NaN handling away and NaN results on real GPUs are not guaranteed. CultMath defines the CPU result by dxc's lowering and the DXIL operation spec (DirectXShaderCompiler docs/DXIL.rst).

NaN does not reliably propagate through CultMath, just as it does not on the GPU. Because min, max, and saturate return the non-NaN operand, some functions turn NaN or infinity into numbers: smoothstep(a, a, a) is 0; infinite inputs to smoothstep and the intercept helpers give numbers; the bezier, smoothstep01, and smootherstep functions return finite values for NaN input. Callers that must detect bad data should check isnan before calling these functions.

Deferred until a consumer needs them: float4x4, transpose, determinant, and HLSL's one-based _11.._33 element names.

HLSL Target: Source Transformations

HlslSourceCompatibilityTests compiles the real shaders/CultMath.hlsl with Roslyn against CultMath. These transformations are the complete list; if a body needs anything else, the test fails:

1. Include guards (#ifndef/#define/#endif) are dropped, except the capacity macro (#ifndef CULTMATH_MAX_SITES, #define CULTMATH_MAX_SITES 128, #endif), which becomes const int CULTMATH_MAX_SITES = 128;.
2. Functions taking Texture2D/SamplerState are dropped; GPU resources have no CultMath analog.
3. File-scope static const becomes const.
4. Floating literals gain f (0.5 becomes 0.5f).
5. Struct fields gain public. HLSL struct members have no access-modifier concept; a C# struct's fields default to private, which would hide them from the mirror test's reflection-based field comparison.
6. An array parameter T name[CULTMATH_MAX_SITES] becomes T[] name.
7. The file body is wrapped in a class, since C# has no free functions; HLSL functions become private instance methods.

HLSL constructs that stay outside the target and must not appear in the mirror: const locals of vector type (C# const is primitives only; write a plain local) and writes through a swizzle component (v.xy.x = s, which C# rejects as modifying a temporary).

The HLSL side is checked too: tools/get-dxc.ps1 fetches Microsoft's DirectXShaderCompiler (x64), and tools/compile-hlsl.ps1 compiles both copies of CultMath.hlsl with it (dxc -T lib_6_3 -HV 2021 -WX, so a warning fails like an error and the text must compile cleanly); the script throws when a shader file is missing, when dxc cannot launch or when it exits non-zero, so a stale exit code never reads as a pass.

The mirror test compares bit patterns (any NaN equals any NaN; -0 differs from 0). It proves the text of CultMath.hlsl computes the same float32 results as C# math on the CPU, not that a GPU agrees: driver sin precision alone makes cultmath_hash and value noise differ bit for bit on hardware. The mirror is compiled under using static CultMath.math;, so its intrinsics are C# math itself: the test proves the composition in the text, not the intrinsic rules, which only HlslSemanticsTests pins.

Every mirror function is compared, internals included: its counterpart is the math function of the same name, public or private (snoise_mod289 and snoise_permute are private), and a function with no counterpart fails by name. A struct parameter maps to the CultMath type of the same name, and only if both have the same instance fields by name, type and order (SpanPath, SpanStage); the argument is built from the case's scalars field by field and copied to its C# twin by field name.

The comparison also allows exactly one struct return shape: a mirror function whose HLSL return type is a plain struct of float/floatN/int/intN fields is compared to its C# counterpart field by field, recursing into each field's own components, down to bit-for-bit scalars, matching each leaf's declaration path (field names, not just position) and declared type as well as its value. CultCellular (math.cs, cellular), CultPhasor (math.Phacelle.cs, phacelle), SpanNoise (math.Span.cs, span_fbm) and CultSiteCells (math.Sites.cs, site_cells_insert) are its consumers. CultSiteCells is also a struct parameter (site_cells_insert, site_cells_border). A struct return with a mismatched, reordered-by-name, or retyped field does not get a second comparison path or a shape of its own; it fails the same walk that already handles a bare vector return.

GLSL Target: Source Transformations

shaders/CultMath.glsl is a GLSL ES 3.00 (WebGL2) library generated from shaders/CultMath.hlsl by GlslLowering (in the test project), and shaders/CultMath.Phacelle.glsl is generated beside it from shaders/CultMath.Phacelle.hlsl. Both are derived files: never edit them by hand. Regenerate them, and the golden fixture tests/CultMath.Tests/fixtures/glsl-parity.json, with:

$env:CULTMATH_WRITE_GLSL = "1"; dotnet test packages/cultmath/tests/CultMath.Tests --filter GlslMirrorTests

Run it on Yggdrasil's mcr.microsoft.com/dotnet/sdk:10.0 image (Ubuntu 24.04) only: that image owns the fixture's platform families and device-bounds-lavapipe.txt, and CULTMATH_WRITE_GLSL=1 fails on any other OS. Re-record the bounds there in the same revision (WRITE_BOUNDS=1, tools/device/run-lavapipe.sh).

GlslMirrorTests.CommittedGlslEqualsLowering fails on any difference between either committed file and the lowering of the committed HLSL. These transformations, in order, are the complete list:

1. Each #include "name" is replaced in place by the included file, by the same resolver the C# mirror uses; GLSL has no #include. The exception is an include under a file-level licence of its own, listed in one table, GlslLowering.SeparateFiles: its line is dropped and the file is lowered on its own, by the same steps, into its own output. The table has one entry, CultMath.Phacelle.hlsl to CultMath.Phacelle.glsl, because Phacelle is MPL-2.0 and MPL-2.0 is file-level: kept separate, it leaves CultMath.glsl MIT. Nothing else in the lowering is keyed by name.
2. Include guards are dropped, and so are the functions taking Texture2D/SamplerState; the host shader samples its own textures.
3. Type names: floatN, intN, uintN and boolN become vecN, ivecN, uvecN and bvecN.
4. File-scope static const becomes const.
5. C-style casts (float)x, (int)x and (uint)x become float(x), int(x) and uint(x). The operand is found by a balanced-bracket scanner: (int)((state >> 28) + 4u) nests parentheses.
6. Intrinsics are renamed: lerp to mix, frac to fract, rsqrt to inversesqrt, asuint to floatBitsToUint, asfloat to uintBitsToFloat, and saturate(x) becomes clamp(x, 0.0, 1.0).
7. Float literals pass through unchanged.
8. Each output is wrapped in a guard named for its file: CULTMATH_GLSL, CULTMATH_PHACELLE_GLSL.
9. The first line names the file as generated, its sources and its licence, and says how to regenerate it; CultMath.glsl's also says which functions live in the separate file.

GLSL ES 3.00 is stricter than HLSL: no implicit conversions between int, uint and float, no C-style casts, and no step(genType, float) overload. Those gaps are closed in the HLSL, never in the lowering: the HLSL is written in the subset both languages share, with edits that are legal HLSL and compute the same bits (step(h, float4(0.0, 0.0, 0.0, 0.0)) rather than step(h, 0.0), + 4u where a uint is added, (float)((uint)hash.x >> 8) before a float multiply). The C# mirror and the FXC and dxc compiles keep those edits honest. A body that needs a transformation outside the list fails GlslMirrorTests.NoHlslOnlyTokensSurvive rather than gaining a special case.

The library declares no precision. The host shader does, and it must be precision highp float; precision highp int;: the PCG hashes need 32-bit integers. tools/compile-glsl.ps1 writes such a wrapper (its own #version 300 es line, the precision, the library by string concatenation, and one call of every public function) and compiles it with glslang, which tools/get-glslang.ps1 fetches pinned. It runs twice: over CultMath.glsl alone, and with -Phacelle over CultMath.glsl followed by CultMath.Phacelle.glsl, the order a consumer that calls cultmath_phacelle concatenates them in. Two negative controls must fail: -Body with an HLSL token (float x = lerp(0.0, 1.0, 0.5);), and -Body calling cultmath_phacelle without -Phacelle, which proves the MIT file does not carry it. The Unity package ships neither GLSL file.

Parity runs in two legs, each with one owner: HLSL against C# by the mirror test, bit for bit; GLSL against C# by the golden fixture, which records 256 seeded cases per function family as float32 bit patterns, evaluated by the consumer on WebGL2 and read back. The families pcg3d and pcg4d are marked exact; the float families are ulp-bounded, the bound measured on the device. A conforming leg runs every fixture family through both routes on a device: tools/device/run-lavapipe.sh compiles the HLSL with dxc and the committed GLSL with glslang, dispatches each family on Mesa lavapipe, and GlslMirrorTests.DeviceResultsHoldFixture judges the results. Exact and integer families must match bit for bit, families with a check pass their Verify, the sampled-form families are checked on the device's own noise values, and every other family is held to the worst error recorded for that device in tests/CultMath.Tests/fixtures/device-bounds-lavapipe.txt.

iv_frustum_ball is ulp-bounded, not exact: it calls sqrt, which GLSL ES 3.00 does not require to be correctly rounded, and WebGL2 compilers may reassociate. On a GTX 1070 under ANGLE's D3D11 backend its radius differs from C# by up to 2 ulp in 43 of 256 cases, sometimes smaller. What the march needs is enclosure, and what culling needs is a ball no wider than its rounding widening, so the family's fixture entry carries a check the consumer applies to every case: from the case's arguments (mx, my, z0, z1, fp, warp), compute in double zm = (z0 + z1) / 2, the centre c = (mx zm, my zm, zm), its |c|_1 = |mx zm| + |my zm| + |zm|, the radius r = (z1 - z0) / 2 sqrt(mx^2 + my^2 + 1) + z1 fp + warp and d = |GPU centre - c|, and require r + d <= GPU radius <= r + d + 2^-19 (r + |c|_1). The lower side is enclosure; the upper side is the bound NoiseBoundTests pins in C#, so a lowering that inflates the ball fails too. Only those two bounds fail the check; the largest ulp distance is reported as for any ulp-bounded family. GoldenFixtureMatchesCSharp applies the same check to C#'s own results.

Every af_* function is an ulp-bounded family, and af_snoise, af_fbm and af_frustum_ball also carry a two-sided check of their own, applied the same way: the device's own snoise or fBm at 75 sampled points of the case's set must lie in the device's form with no tolerance (a device whose gradient error exceeds the allowance fails), its width must not pass the Lipschitz form's allowance, and the device's af_frustum_ball radius must enclose the sampled ray points and stay within its rounding widening. The reference is the device's own noise, which is right for culling soundness but would pass a device whose noise is wrong everywhere, so two appended families (af_snoise and af_fbm, 256 cases each) add the centre leg: the fixture's own x0 (C#'s value at the centre) must lie in the device form's [x0 - e, x0 + e], for every af_snoise and af_fbm family. They also draw reach |axis| + radius from 1e-6 to 1e-3, the short slices near a camera, where the float32 allowance is most of e: the main families' reach is at least about 1e-3, where the second-order remainder is far larger and a device that drops the allowance still passes. GoldenFixtureMatchesCSharp requires each of the two to hold cases that fail on C#'s own results once the allowance is removed.

Four more appended families, the coordinate-band snoise_grad and snoise_hessian reads (256 cases each, the magnitude of the point log-uniform from 1e-4 to 1e2 and from 1e2 to 4e6), carry an allowance check: with S = |x|_1 and the allowances math.span_slack(math.span_rounding(1, x, zero stage, 0, 0, 0)) (the owners of the span stage's allowances in C#, HLSL and GLSL), the result must lie within the allowance of the kernel's exact value, gradient or Hessian in double, no tolerance. The two small families must also fail it in at least 16 cases once the absolute term is removed. GoldenFixtureMatchesCSharp applies the check to C#'s own results, and a device capture is judged by the same Verify.

GoldenFixtureMatchesCSharp evaluates C# on the committed fixture's own arguments. Arguments are drawn only when the fixture is regenerated (CULTMATH_WRITE_GLSL=1), because the draw itself calls MathF.Pow, whose bits differ by OS. iv_exp and phacelle take their C# results from the platform's exp, sin and cos, which also differ by OS, so their entries name the platform that generated them and the test compares them case by case against the running platform, printing the largest distance. iv_exp ("ulp-bounded, platform exp") must agree within one ulp. phacelle ("ulp-bounded, platform exp, sin, cos; across platforms |diff| <= 2^-20 max(|v|, 1); on a device |diff| <= 2^-18 max(|v|, 1)") must agree across platforms within 2^-20 max(|v|, 1) per component, v being the fixture's value: its sums of exp, sin and cos cancel near zero, where ulps mean nothing, and Windows differs from the Linux fixture by up to 3.6e-7 max(|v|, 1). The device clause is 2^-18 because the devices measured are farther: lavapipe is 2.2e-6 max(|v|, 1) and the GTX 1070 3.1e-6; the conforming run judges it. Every other family is compared as text. phacelle is in CultMath.Phacelle.glsl, so the evaluator concatenates CultMath.Phacelle.glsl after CultMath.glsl for that family, with normalization 0.5. The site does not vendor the MPL file and skips the family. Its evaluator is the lavapipe GLSL route of the conforming run above, which covers every fixture family, phacelle included. The real-device leg (ANGLE D3D11 on the GTX 1070) is the later cut angle-capture; the one GLSL evaluation of phacelle on that device is a single probe (GTX 1070, ANGLE D3D11: within 3.1e-6 max(|v|, 1) of C#). Asura's gpu-snoise-parity is a different check, an FXC and Unity readback of snoise, and evaluates no GLSL.

Invariant 8: Value-and-Gradient Primitives

A primitive tagged invariant 8 returns its value and analytic gradient together, laid out as float4(gradient.xyz, value.w), with no value-only twin (a value-only form would be a second path with no consumer). smin_grad and cellular are the first primitives in this family:

• smin_grad(float4 a, float4 b, float k) is Inigo Quilez's quadratic-polynomial smooth minimum ("smooth minimum", https://iquilezles.org/articles/smin/), carried to value-and-gradient form. Its blend factor h is affine in b.w - a.w, so differentiating the value with respect to position, the terms carrying dh/dp cancel exactly; the surviving gradient is lerp(∇b, ∇a, h), the same h the value uses. This is the analytic gradient, not an approximation, and it is continuous across the |a.w - b.w| = k seam where h saturates to 0 or 1.

• cellular(float3 p) is Worley's cellular texture basis function ("A Cellular Texture Basis Function", SIGGRAPH 1996), searched over the jittered 5×5×5 neighbourhood of feature points selected by pcg3d (never the sin-based hash), with cells whose position-only lower bound already exceeds the running F2 skipped before their hash is even computed. It returns CultCellular { nearest, edge, id }: nearest is (∇F1, F1), edge is (∇F2 - ∇F1, F2 - F1), and id is the nearest cell's pcg4d hash (over the cell coordinate alone, never the pcg3d hash that produced its jitter) mapped to [0, 1). ∇F1 and ∇F2 are each undefined exactly at their own feature point (F1 = 0 or F2 = 0); cellular follows the repo's existing degenerate-normal convention (GameCult.Geometry.CultGeometryIsoSurface.EmitOrientedTriangle, which guards a zero-length normal and returns the zero vector instead of the NaN a bare normalize gives there) and returns the zero vector rather than NaN for either. F1 = 0 is ordinary (p sits on a feature point); F2 = 0 needs two distinct cells' pcg3d-jittered feature points to land on the same float32 value in every component, which is unreachable at ordinary magnitudes but confirmed reachable (see "Precision domain" below). The identity of the nearest and second-nearest feature point changes discontinuously across the F1 = F2 set (a genuine kink, not a numerical artifact), so ∇F1 and ∇F2 are only continuous away from that set.

  The 5×5×5 radius is exact, not merely safer than the classic 3×3×3 radius: for a query at position u inside its own cell (u in [0, 1)^3 per component) and a candidate feature at integer cell offset d with jitter j in [0, 1)^3, the per-axis displacement (u_i - j_i) - d_i is always strictly greater in magnitude than max(0, |d_i| - 1), because u_i - j_i is always strictly inside (-1, 1). So any cell with |d_i| >= 3 on some axis is strictly farther than 2 from the query. Two real, always-present candidates bound F1 and F2: the query's own cell, whose distance is always strictly less than sqrt(3) (every axis strictly under 1); and the near-side neighbour on whichever single axis has u_i closest to a cell boundary. Let m = min(u_i, 1 - u_i) on that chosen axis, so m is in [0, 1/2] by construction. Stepping one cell towards that nearer boundary puts the query within 1 + m of the far face on the offset axis and within 1 - m of the near face on each of the other two axes, so that neighbour's squared distance is at most (1 + m)^2 + 2*(1 - m)^2 = 3 - 2m + 3m^2 <= 3 for every m in [0, 1/2] (3 at m = 0, dipping to a minimum of 8/3 ~= 2.667 at m = 1/3, then rising back to 2.75 at m = 1/2), and strictly below 3 because the jitter never reaches exactly 0 or 1. F2 is at most the larger of these two real candidates (any two real candidates upper-bound the 2nd-smallest value over the full infinite candidate set), so F1 < sqrt(3) and F2 < sqrt(3) always, comfortably under the radius-2 exclusion distance of 2, with margin 2 - sqrt(3) ~= 0.268 (Soul's adversarial check, cut 2a-i: the worst realizable in-box F2 is exactly sqrt(3), reached only in the unattainable limit at a cell corner). 3×3×3's matching argument only reaches radius 1, whose excluded cells start at distance 1, short of the sqrt(3) ceiling — the gap its measured failures (9 of 400,000 points in [-50, 50]^3, worst error 0.088) live in.

  An earlier version of this proof picked the near-side neighbour on a fixed axis instead of whichever axis sits closest to its own boundary, and that version is wrong (Soul, cut 2a-i): u = (0.5, 0, 0) with the x-neighbour's jittered feature at (-1, 1 - 2^-24, 1 - 2^-24) gives a distance of about 2.06, outside the radius-2 exclusion the proof needs — u.x = 0.5 is the point on the cell farthest from any boundary, so x was the wrong axis to fix there. Choosing the axis by proximity to its own boundary is what keeps m <= 1/2 and the bound at sqrt(3).

  CellularAndSminGradTests pins both the prune's bit-identical equivalence to an unpruned 5×5×5 reference and the search's exact agreement with an independent 7×7×7 brute-force oracle.

  Precision domain: cellular's exactness above is a statement about the search radius, not about float32. Past |p| of roughly 2^23, adjacent integer cells stop being distinguishable from their neighbours at float32 precision in some directions, and distinct cells' jittered features can round onto the same value, making F1 = F2 ties (and, per-component, F2 = 0) increasingly common from about 1e6 and confirmed at 2^24 (measured: 4096 of 4096 sampled integer points at 2^24 hit this before the guard above). cellular stays finite there; it does not stay accurate in the sense of matching the idealized real-valued Worley field.

  The lower-bound prune's bit-identity to the unpruned 5×5×5 search also has a domain: it holds for |p| < 2^25. Above that, cell + d (an integer cell coordinate plus a small integer offset) starts to round in float32, so a pruned cell's own recomputed offset can land on a different neighbour than the unpruned loop visits at that same nominal (dx, dy, dz), and the two search paths diverge (Soul, cut 2a-i: measured 180 of 1,100,000 points differ at |p| ~= 3.4e7). This is a float32-precision limit on the prune's equivalence proof, not a bug in the prune itself: below 2^25 the two paths are proven bit-identical (PrunedSearchIsBitIdenticalToTheUnprunedFiveCubedSearch, which samples [-50, 50]^3, is comfortably inside that domain).

• snoise(float3) follows webgl-noise src/noise3D.glsl as of stegu/webgl-noise 22434e04d7: kernel radius squared 0.5, output scale 105, permutation mod289((34x+10)x). Upstream's 21d9fe23d7 (2020-10-14, "an age-old bug that caused slight discontinuities along simplex boundaries") moved the radius from 0.6 to 0.5; at 0.6 a lattice vertex outside the four summed corners can lie inside the kernel, leaving a value jump at the cell boundary (3.1e-3 over the committed probe's 400 lines). ff3b5d34ea set the scale to 105 and a3e6d57095 (2021-06-30) the permutation; the 2D snoise shares that permutation helper, so its values changed too, while its own kernel (0.5) and scale (130) already matched upstream. NoiseGradTests.SnoiseAndItsGradientAreContinuousAcrossSimplexCellBoundaries pins the continuity. Ties in the offset x0 resolve in Gustavson's simplexnoise1234 order (x over y, y over z, x over z), a total order, so the four corners are always a simplex. Upstream's step compares are cyclic: where the three components are equal, which is every input whose differences y - x and z - x are whole numbers (every integer point, the origin and every (t, t, t) among them), the corners repeat the origin vertex and the value jumps by about 0.77 near the origin and up to 1.56 at magnitudes of 64 and above. SnoiseTieScopeTests pins the set against the 0.3.0 values. NoiseGradTests.SnoiseIsContinuousOnTheSimplexDiagonal pins the order.

• snoise_grad(float3 p) carries snoise(float3) to value-and-gradient form, following the differentiation in webgl-noise src/noise3Dgrad.glsl (Ashima Arts / Ian McEwan, MIT), which uses the same radius, permutation and scale as snoise(float3): per corner, m0 = max(0.5 - dot(x,x), 0), so d(m0^4 * dot(p,x))/dx = -8*m0^3*dot(p,x)*x + m0^4*p, summed over the four corners and scaled by the same constant as the value, so .w is exactly snoise(float3). fbm_grad and ridged_grad are octave sums of snoise_grad: each octave samples at p * frequency, so by the chain rule its gradient scales by that same frequency, and both amplitude (gain) and frequency (lacunarity) compound per octave rather than applying once to the whole sum. ridged_grad folds each octave about zero (Σ aᵢ(1 − |nᵢ|), gradient −Σ aᵢfᵢ sign(nᵢ)∇nᵢ, Musgrave's ridged multifractal, Texturing and Modeling ch. 16); the crease at nᵢ = 0 is deliberate (the operator's dune term, 2026-09-25), not a bug. Both clamp octaves to [0, 16]: HlslSourceCompatibilityTests's bit-parity oracle drives every mirrored int parameter across the full int32 range, which is the right domain for pcg3d/pcg4d's O(1) hash inputs but would turn an unclamped octave count into a multi-billion-iteration loop for these two mirrors.

• phacelle(float3 p, float3 side, float offset, float normalization) is Rune Skovbo Johansen's Phacelle noise ("phase + cell": a stripe pattern built by blending one cosine/sine wave per jittered cell), generalised to 3D cells. It returns CultPhasor { cosine, sine }, each (∇, value). side is the stripe wave vector, whose length is 2π times the stripe frequency; in 3D the perpendicular to the flow is not unique, so the caller chooses it (on a unit sphere, cross(p̂, flow)). offset is the phase in cycles. It visits 4×4×4 cells with pcg3d-hashed jitter in [-0.5, 0.5), weights each wave by max(0, exp(-2d²) - 0.01111), and normalizes the blended phasor as upstream does (length 1 where the raw length is at least 1 - normalization, scaled by 1 / (1 - normalization) below). The gradient is exact, weight derivatives and normalization included, rather than upstream's ∇cos ≈ -sin·side, which ignores the weight gradient. It is discontinuous where the raw length crosses the normalization threshold and where a cell's weight leaves its support (d² = 2.24995, distance 1.49998); the value is continuous at both. At the support edge ∇w jumps from -4·0.01111·v to 0, a per-cell jump of length 0.0444·|v| (0.0667 at |v| = 1.5); measured over random points, the output gradient jumps by a median 1.5% of its length and at most about 27%. normalization is meant to lie in [0, 1]: 1 or more saturates (every output is stretched to length 1), a negative value scales every output down by 1 / (1 - normalization), and the normalized gradient scales as 1 / |I| for the blended phasor I, so it grows where the blend nearly cancels. The total weight W is positive for every pcg3d input (the smallest found by hill-climbing is 0.056); W = 0, and a NaN output, needs adversarial jitter no hash yields. Cells whose per-axis lower bound Σ max(0, |local - g| - 0.5)² is at least 2.25 lie outside the support and have weight exactly 0, so they are skipped, which changes no output bit. Cells whose bound sits just under 2.25 can still carry a nonzero weight, so the cut cannot be lowered; PhacelleTests pins the cut with points next to such cells, against an internal overload with the prune disabled, and the HLSL mirror test pins it on the shader side. The prune keeps about 45 of the 64 cells on average (a saving of about 30%); about 14 have a nonzero weight. The port sits in its own MPL-2.0 files (math.Phacelle.cs, CultPhasor.cs, CultMath.Phacelle.hlsl); see THIRD-PARTY-NOTICES.md. No 2D variant exists; nothing consumes one.

Integer hashing uses the PCG hashes from Jarzynski and Olano, "Hash Functions for GPU Rendering" (JCGT 9(3), 2020): pcg(uint) is O'Neill's RXS-M-XS 32/32 permutation over one LCG step, and pcg3d/pcg4d are the paper's (3 → 3) and (4 → 4) hashes. They use only uint multiply, add, xor, and shifts. dxc wraps uint multiply and add modulo 2^32 exactly as C# unchecked arithmetic does, so they are integer-exact in HLSL and on the CPU, GPU included, unlike the sin-based hash. CultMath has no uint vectors, so pcg3d(int3) and pcg4d(int4) carry uint bit patterns in int components. The float2/float3/float4 overloads hash the IEEE-754 bits (asuint), so -0 and 0 hash differently; they exist for callers that already hold a hash key as float bits, and pcg3d(float2) holds z at 0, which is the paper's route for (2 → N) inputs. Seeds and other scalar uses take one output component.

cellular's jitter and id always hash the integer cell coordinate directly (pcg3d(int3(neighbor)), pcg4d(int4(int3(cellId), 0))), never a cell coordinate's float32 bit pattern. Hashing the bit pattern of an integer-valued float is not the same permutation as hashing the integer: two adjacent cells' bit patterns share more structure (both have the same exponent and a mantissa that differs by one increment) than two adjacent integers do going into pcg3d's own mixing, and that residual structure survived into the jitter. Hashing the integer cell coordinate directly is exact and runtime-independent, and it is the HLSL mirror's own hash, so C# and shader agree bit-for-bit. What pins this is the oracle and mirror tests, not a uniformity measurement: reverting to hashing the float bits (or the sin-based hash) is caught by ProductionSearchMatchesTheBruteForceOracleExactly, EveryMirrorFunctionMatchesCSharpMath, F1NeverExceedsF2, and the other tests that key off the integer-hash fixtures (cut 2a-i). CellularAndSminGradTests (JitterAndIdPassAChiSquareUniformityTest) checks the uniformity of the hash as cellular uses it: the jitter marginals and the id pass a chi-square test at a stated significance against pcg3d(int3(...)).

Where HLSL is silent, CultMath keeps its own decisions and does not defer to Unity.Mathematics: hash returns float, and Random is CultMath's own xorshift32. Engine-shaped helpers that HLSL lacks (float2x2.Rotate, float3x3.Euler, quaternion, snoise) enter only when a consumer needs them, with their semantics stated in code.

Known C# friction: under using static CultMath.math; the constructor functions (float3(...), float3x3(...)) hide the type names in member access, so static members need a qualified type (CultMath.float3x3.Euler(...)).

Sites

A site is float4(position.xyz, weight); a 2D consumer uses z = 0. math.Sites.cs scores a query point against a site two ways and keeps the two best:

• power_score(p, site): value |p - s|^2 - w^2, gradient 2 (p - s). A power cell is the intersection of the half-spaces bounded by the radical planes against every other site, so power_border is exact.
• multiplicative_score(p, site): value |p - s| / w, gradient (p - s) / (|p - s| w), zero at p = s (cellular's degenerate convention). Its borders are Apollonius circles, and the border distance is first order.

A span function is a C# loop over a per-site step that has a shader twin, because the HLSL mirror cannot take a span or a buffer. site_cells_empty() starts the fold, site_cells_insert(cells, score, index) folds one score in (strict < against F1, the old F1 moving to second, else strict < against F2, so an exact tie keeps the earlier index first), and site_cells_border(cells) reads float4(unit normal, distance), with n = grad F2 - grad F1 pointing toward the nearest site and (0, 0, 0, 1.0e30) when there is no second site. That border is the distance to the bisector of the two lowest scores, first order and for either metric; it is not the distance to the nearest site's cell boundary (the F2 - F1 edge error: the second-lowest score is not the nearest plane in general). The exact power cell border is a second pass: power_border_step(p, owner, other, border) takes the running float4(unit normal toward the owner, distance) and the radical plane of owner against other, with the distance built from the offset owner - p and the weight difference (w1 - w2)(w1 + w2) so it does not cancel for close sites seen from far; the cell boundary distance is the least over every site except the owner (the fold's nearest), and a site coincident with the owner has no plane. The step trusts the caller's owner: a query outside the owner's cell reports distance 0 with that plane's normal, not an error. power_border(p, sites) is that loop; the shader's cultmath_power_border(p, sites, count) is the same loop over a by-value array (see "Sites in a shader" below). Cell spans refuse a non-finite query, site position or weight, and more than 2^24 sites (indices are floats). power_cells and multiplicative_cells are that loop over a ReadOnlySpan<float4> and nothing else; a shader caller writes the same loop over its own array. Every site of an explicit span is visited, so no window and no weight-ratio range apply. CultSiteCells holds the two score fields and the two site indices (floats, -1 for none).

gabriel_margin(a, b, other) is (|other - (a + b) / 2| - r) / r with r = |a - b| / 2: zero on the diameter circle, negative inside it, -1 when a and b coincide. soft_gabriel_gate(sites, first, second, softness) is smoothstep(-softness, softness, least margin over every other site) on positions alone, 1 when no other site exists; it is C# only. distance_to_segment(point, start, end) has out-free 2D and 3D forms beside the out form, all over one public segment_closest per dimension, each with its 2D and 3D shader overload.

Weighted lattice cells

weighted_lattice_cells(p, time, bodies, cellSize, drift, boost, falloff, smoothing, stretchNormal, stretch) in math.Lattice.cs returns float4(F1, F2, smooth, id) over a jittered, drifting lattice, in cell units times weight. Four public steps compose it, and a shader or a tape composes the same four:

• lattice_site(cell, time, drift): the site cell + 0.5 + 0.5 sin(drift time + 2 pi u) per axis (so it stays within one cell of the corner) and the base weight 0.5 + 0.5 u, in [0.5, 1), from its own hash lane.
• lattice_weight(base, border, boost, falloff): base (1 + boost exp(-(border / falloff)^2)). border is power_border at the cell centre over the body span (exact, so the weight is continuous as bodies move; the first order site_cells_border jumps where the second-nearest body changes and is not used).
• lattice_distance(q, site, weight, normal, stretch): weight |(stretch a, c)|, a and c the offset along and across the unit normal; isotropic at stretch 1.
• lattice_smooth_step(m, d, f1, smoothing): the smooth minimum fold, started at F1 + smoothing and clamped to it, so a cell with d >= F1 + 2 smoothing changes nothing and the result does not depend on the box visited.

The search visits Chebyshev rings around the query's cell and stops before ring k once 0.5 (k - 1) >= max(F2, F1 + 2 smoothing): every cell on ring k has d >= 0.5 (k - 1) (site within one cell of its corner, weight at least 0.5, stretch at least 1). A second pass folds the visited box in absolute row-major order, so ties and the smooth minimum do not depend on how far the search went. The proof is the comment above weighted_lattice_cells.

The accepted range is boost 0 to LatticeMaxBoost (4, a weight ratio of 10:1), stretch 1 to LatticeMaxStretch (2) and smoothing up to LatticeMaxSmoothing (1); outside it the function throws ArgumentException. LatticeMaxRing (37) is the proven greatest ring at the range edge. Measured over 24,000 points on orbiting bodies (six bodies, sigma ratio 6:1, the ruled look and the range edge), the greatest ring visited has median 3, 99th percentile 7 and maximum 8 (49, 225 and 289 cells); RingsStayWellInsideTheCap asserts these.

Sites in a shader

A shader function cannot take a span, so cultmath_power_border(p, sites, count), cultmath_lattice_cell_distance(...) and cultmath_weighted_lattice_cells(p, time, bodies, count, ...) take float4 sites[CULTMATH_MAX_SITES] by value and the count in use, with count in [0, CULTMATH_MAX_SITES]. CULTMATH_MAX_SITES is 128 unless the consumer defines it before including the library (#ifndef): 128 covers every Aetheria zone within GLSL ES 3.00's 224 guaranteed fragment uniform vectors, and loops run to count, so the capacity costs uniform space, not iterations. A caller whose sites live in a buffer copies them into a local array first; the library takes no resource, accessor macro or uniform of its own. The preconditions the C# functions throw for are the caller's in a shader (finite arguments, cellSize > 0, the stated ranges, p within 2^23 cells). The ring pass is bounded k <= 37 and exits as C# does; the box pass recomputes each distance instead of caching 75 x 75 of them, which gives the same values. The C# mirror compiles the array as T[], and HlslSourceCompatibilityTests.SiteArrayMirrorsMatchCSharp compares the three functions through the span twins in SiteArrays (reflection cannot pass a ReadOnlySpan); CULTMATH_MAX_SITES is a shader-only capacity with no C# twin.

Intervals

math.Interval.cs and shaders/CultMath.Interval.hlsl (included from CultMath.hlsl after the Phacelle include) carry interval arithmetic for culling empty space with a proof. Their invariant, intervals-enclose: every interval function returns a [lo, hi] that contains f(x) for every x in its input region, where f(x) is the same function evaluated pointwise in float32. A consumer that skips a region because hi < cutoff has skipped nothing that was there.

Representation: an interval is float2(lo, hi) with lo <= hi, both finite. There is no empty interval; a caller that needs "provably empty" tests hi < cutoff. iv_point(x) is float2(x, x).

Ops: iv_point, iv_add, iv_sub, iv_neg, iv_mul (the least and greatest of the four corner products), iv_scale(iv, float), iv_abs and iv_sqr (an interval straddling zero starts at 0), iv_min, iv_max, iv_sqrt (lo clamped at 0), iv_exp, iv_clamp(iv, float, float), iv_saturate, iv_smoothstep(float, float, iv) (its ordered endpoints) and iv_lerp(iv, iv, float). No others until a consumer names one. Each has an HLSL mirror named cultmath_iv_*, compared bit for bit by HlslSourceCompatibilityTests.

The ulp-widening rule. Each bound is computed with the pointwise function's own float32 operations. IEEE round-to-nearest is monotone, so an op that is monotone on its region encloses its float32 evaluation exactly, with no widening; that covers every op above except two, which widen by exactly one ulp, in the implementation and never in the test:

• iv_exp: exp is not required to be correctly rounded, so a libm need not be monotone. .NET's MathF.Exp on Linux steps down 0 times over every float in [-87, 88.7] (IntervalTests.MeasureMonotonicity), so on that CPU the widening is a margin; it is there for other libms and GPUs.
• iv_smoothstep: t * t * (3 - 2t) in float32 steps down 337,095 times over the floats in [0, 1], each time by at most 1 ulp (MeasureMonotonicity). Since t = saturate((x - minimum) / (maximum - minimum)) is itself monotone in x, that covers every smoothstep.

iv_lerp is the natural interval extension of x + (y - x) * amount, which is how lerp evaluates; every step is monotone, so it encloses with no widening for any amount, and it is tight only when a is a point, because x appears twice.

IntervalTests runs every op over 10,000 seeded intervals (centres in [-20, 20], widths log-uniform in [1e-4, 10], every eighth a point) and 16 points inside each, endpoints and zero included, with no tolerance; and for every op it raises lo and lowers hi by one ulp and requires the harness to catch each. A new op is one entry in that table.

Noise bounds. iv_snoise_ball(c, r) is [n - L r, n + L r] intersected with [-1, 1], where n = snoise(c) and L = SNOISE_LIPSCHITZ: one snoise, value only. No interval function reads snoise_grad, and none has a value-only or gradient twin. iv_fbm_ball(c, r, octaves, lacunarity, gain) is the octave sum of iv_snoise_ball(c * f_i, r * |f_i|) scaled by a_i, compounding frequency and amplitude exactly as fbm_grad does, octaves clamped to [0, 16]. A domain warp that moves a point by at most D is enclosed by the radius r + D (NoiseBoundTests.WarpedPointsStayEnclosed); this is what the gamecult.org ground's march rests on.

iv_frustum_ball(m_c, z0, z1, footprintPerDepth, warp) is the ball one probe serves a whole screen tile with. In a pinhole camera frame looking down +z, where the ray of slope m is the points (m z, z), it encloses every point of every ray whose slope lies within footprintPerDepth of m_c (N / (sqrt(2) f) for an N x N tile at focal length f pixels: the full pixel footprint, so sub-pixel jitter is covered), over depths [z0, z1], each point moved by a warp of length at most warp. The centre is (m_c z_m, z_m), z_m the mid-depth; the radius is ((z1 - z0) / 2) |(m_c, 1)| + z1 footprintPerDepth + warp. A point of slope m_c + e at depth z is the centre plus (z - z_m)(m_c, 1) plus z (e, 0). The returned ball is float32, and on a degenerate (z0 == z1), thin or far segment the rounding of the centre alone exceeds the gap the triangle inequality leaves, so the radius is widened by (|c|_1 + radius) 2^-20. With u = 2^-24, each centre component carries at most two roundings (z_m, then the product), so the float centre is within 2u |c|_1 of the exact one; the radius carries at most seven (z1 - z0, the axis length at 3u, the product, z1 footprintPerDepth, the sum, + warp), so the exact radius is at most (1 + 7u) times the float one; the widening rounds once more. 2^-20 = 16u covers 8u (|c|_1 + radius) twice over, and fma contraction only removes roundings. The axis count assumes a correctly rounded sqrt, which IEEE gives C#. GLSL ES 3.00 does not: it inherits sqrt's precision from inversesqrt, and D3D-backed compilers may reassociate. So the bound is proven for C# and HLSL on IEEE hardware, and on GLSL it is measured: on a GTX 1070 under ANGLE's D3D11 backend the fixture's 256 cases keep a minimum relative slack of 1.98e-6 (about 33u), the same as C#'s. The fixture's enclosure check is what measures it on each device. The caller rotates and translates the centre into world space and scales centre and radius by its noise frequency; the rounding of that transform is the caller's. NoiseBoundTests.TileBallEnclosesEveryRaySegment checks 2,000 seeded tiles x 64 points of the r2 domain and 2,000 tiles x 8 corner points of each of seven extreme families (z0 == z1, z1 - z0 = 1e-6 z1, |m_c| to 1000, z to 1e7, z0 = 0 with z1 to 1e-3, f down to 1, warp to 1e6), corner flows full-length and outward, distances in double, with no tolerance. It pins the ball to the derivation both ways (r <= ball.w <= r + 2^-19 (r + |c|_1)), so a ball that forgets the widening fails and one grown past it fails too. A moving camera needs nothing more: each frame's probes use that frame's camera, and no ball, mask or probe result is carried from one frame to the next.

The noise ball alone proves little: at L near 10 it bounds snoise to a width of 2 L r, which covers all of [-1, 1] once L r reaches 1. Empty space becomes provable through an analytic envelope. The consumer's density is a pre-distortion SDF (a height fog, a well map, a carved sphere) that noise displaces only near its surface, and the consumer composes the bound from iv_* ops. The envelope is taken over the slice's exact box (iv_mul of the depth range by each slope range, then the SDF's own monotone pieces), together with its fade. The noise is iv_snoise_ball over iv_frustum_ball, scaled by the noise frequency. The warp moves only the noise argument, so it enlarges the ball and the envelope's box takes none of it. Where the fade's upper end is 0 the noise term is identically 0 and no snoise is evaluated. Where the envelope already proves the slice empty with the noise at its full range, the ball is not evaluated either. A probe is worth attempting only when the best result any centre value could give would prove the slice, and a ball with L r >= 2 is [-1, 1] whatever snoise(c) is, so it costs nothing. CultMath owns none of the envelope: the interval files have no envelope function, and each consumer composes its own. NoiseBoundTests.EnvelopeBoundEnclosesDensity checks that composition, the one the site and Aetheria use, over 2,000 seeded slices x 64 warped points with no tolerance.

L has provenance, not a proof. NoiseBoundTests.MeasureLipschitz (slow, explicit) takes the largest |snoise_grad| over 1e6 seeded points in [-256, 256]^3 (7.1933103), refines the 1e4 largest by gradient ascent (7.2400064), and multiplies by 1.10: L = 7.9640074. It also reaches |snoise| = 0.9718335 by the same ascent on the value, which is what lets the bound intersect with [-1, 1]. LipschitzConstantPinsSampledGradients checks 1e5 more seeded points against L and climbs again from the start MeasureLipschitz prints as its witness, failing if L is not 1.10 times that maximum to within half a percent, so a 1% move of L fails. The enclosure tests over 2,000 balls x 64 points are the defence. A change to the snoise kernel must re-run MeasureLipschitz and re-pin L and the witness.

What the ball bounds guarantee: enclosure of snoise and fbm_grad(...).w over the ball, in float32. What they do not: anything about tapes, pruning or choice tracking. Correlation between the operands of one expression is what "Affine forms" below adds. How loose the ball bounds are is a number: NoiseBoundTests.TightnessReport prints the mean interval width over the sampled range: 5.08 for iv_snoise_ball at r = 0.05, 1.96 at 0.25 and 1.28 at 1.0; 6.64 and 2.82 for 4-octave iv_fbm_ball at 0.05 and 0.25.

The saving is measured where there is empty space to find. NoiseBoundTests.IntervalSkipHalvesEvaluations draws screen tiles, each with its own camera, and runs one pre-pass per tile over the dense grid's cell ranges: it starts with the whole grid, skips and doubles a range it proves empty, halves one it cannot, and runs to the grid's end. It then draws the tile's N^2 ray slopes, one per pixel, jittered inside it. Each ray is marched three times with the same integrator and the same early-out at transmittance 0.02: over every cell (dense), over the tile's unproven cells (the tile march), and over the oracle mask, the cells whose mid-depth density is nonzero for at least one of the tile's rays. The tile march and the oracle march integrate the dense march's nonzero cells, so their transmittance agrees with it exactly in every configuration. The oracle is a ceiling the pre-pass cannot move: dense cost over oracle cost. Against it the test prints the efficiency (oracle cost over the tile march's cost with its probes) and the cull fraction (the oracle-empty cells the pre-pass proves empty). The probe overhead is the tile march's cost over that cost plus the probes'. Cost is counted in snoise evaluations, with an envelope evaluation or probe weighted 0.2. That weight is an estimate, and both counts are printed. The flow is not counted. At N = 8, in Aetheria's units on its 256-cell quadratic grid out to the Main Camera's far plane, 2048:

• (a) Height fog, camera above the safety band, rays level and up: the whole grid is one envelope probe per tile, against 256 envelope evaluations per ray, 16384x. Every cell is oracle-empty and the pre-pass proves them all.
• (b) Inside the fog: 1.16x, and 1.11x with the warp. Most cells are fog, and the pre-pass probes each one.
• (c) The deep well (ruling wells-scale-order-of-magnitude): the zone bowl plus one well at its centre, PowerPulse exponent 2, scale 2220, depth 600. The band G = 60 is where only the noise decides density (s within F -+ 1.5 A with the fade at 1); depth and half-depth radius are each 10 G. The camera's xz is uniform within 300 of the centre, at y = F - h(xz) / 2, so it looks across the bowl from inside it. Both rows run on the same seeds. With no warp, the centred interval takes 3.20x (oracle ceiling 5.24x, efficiency 0.611, cull fraction 0.988, probe overhead 0.940) and the affine bound 3.57x (efficiency 0.682, cull 0.993, overhead 0.905). With the warp D = 60: 3.12x (ceiling 5.31x, efficiency 0.588, cull 0.987, overhead 0.929) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine is 1.12x and 1.11x cheaper. The range mask (the oracle cells plus every cell where any of 5 x 5 slopes x 9 depths has density above zero) gives a range ceiling of 0.859 (0.848 with the warp), the cost a bound that proves exactly the range-empty cells would reach against the oracle's; the efficiencies against it are 0.794 and 0.712 (warp 0), 0.768 and 0.694 (warp D).
• (c-band) The shallow wells, four mass-derived wells in the zone bowl with the camera gazing across, whose geometry is band-dominated, printed only. Centred interval 3.01x (efficiency 0.414, cull 0.960, overhead 0.958), affine 3.95x (0.544, 0.980, 0.901); with the warp 2.75x (0.383, 0.954, 0.944) against 3.54x (0.494, 0.976, 0.865).
• (d) r1's uniform slab, which has no envelope: 1.38x with the warp 0.05, and 1.67x with no warp.
• (e) The void at the shipped point (ruling operator-shipped-void): Rh = 198, S = 50, Rc = Rh + S = 248, e = -ln 1.5 / ln(1 - (Rh / Rc)^2), the shipped camera, 200 tiles. The fixed-step reference takes 36.34 steps per pixel. The same cells masked by the pre-pass take 20.08, with identical transmittance. The footprint-aware march takes 13.50: 2.69x fewer. snoise per pixel goes from 32.98 to 26.70, and combined cost from 40.25 to 29.54 with the probes, 1.36x. Every one of the 200 tiles proves a body. Before the body both marches are measured against a converged one (midpoint quadrature at h = 0.5 over every cell, trusting no pre-pass): the relative optical-depth error at the body start is at most 0.0608 for the fixed-step march and 0.08639 for the footprint march, 1.42x. Over the grid of hollow radius, camera offset and ramp width the steps ratio runs from 1.59x (Rh = 100, low, S = 150) to 27.45x (Rh = 1000, centred, S = 10), the cost ratio from 1.26x to 3.89x, and the depth-error ratio from 0.85 to 4.72 (Rh = 200, centred, S = 150, where the errors are 0.01504 and 0.003188). LOD-far takes 14.19x fewer steps; its depth errors, 0.2849 and 0.205, are not comparable, because its fixed-step reference keeps the octaves the footprint march truncates.

The analytic body (ruling operator-analytic-wall-body). When a single dense cell's own bound is a positive point, the pre-pass spends one more probe over the rest of the grid. If that bound is a positive point too, every point of every ray of the tile from there on has exactly that density, so the footprint march takes one closed-form step, exp(-BodyDepth), from the body start to the grid's end and stops. BodyDepth is density x extinction x length x |dir|. NoiseBoundTests.AnalyticBodyMatchesFineMarch pins it: the bound, the mask and the pointwise densities agree exactly at the body, and BodyDepth equals the summed fixed-step depth of the body cells within 1e-4 relative. The comparison is in the depth domain, so LOD-far's deep bodies do not underflow.

The footprint-truncated void weights its octaves by w_i(z), decreasing in z. Its bound takes each weight over [w_i(z1), 1] rather than [w_i(z1), w_i(z0)], because one bound serves both the full field (w = 1) and the truncated one. [w_i(z1), 1] contains both, and iv_mul is monotone in its interval argument, so the composed interval encloses both fields. NoiseBoundTests.WeightBoundEnclosesOneSignedNoise checks it on LOD-far over 2,000 segments: each weight lies in its bound, and w(z) n lies in iv_mul(bound, n) for a one-signed n. A lower end of w(z0) fails it.

The contracts, at N = 8, the lower of warp 0 and D:

• (a) at least 2x combined cost;
• (c) on the deep well (rulings affine-wells-contract-cheaper-than-interval and wells-overhead-plus-floor): the affine bound's combined cost ratio is above the centred interval's at warp 0 and at warp D; and, hard, the pre-pass proves at least half of the oracle-empty cells empty under both bounds, so a pre-pass that skips nothing fails. The probe overhead, the efficiencies against the oracle and the range ceiling, the ceilings and the cull fraction are printed, not asserted; (c-band) is printed only;
• (e) at the shipped void: the masked fixed-step march agrees with the reference exactly; the footprint march's optical depth at the body start is off the converged march's by at most twice the fixed-step march's error; and it takes at most half the reference's steps per pixel.

This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against the centred interval's 3.20x and 3.12x, with a cull fraction of 0.987 at the lowest, (e) 2.69x fewer steps and a depth error 1.42x the fixed-step march's. The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are not tuned toward the contracts. An affine probe is one snoise_grad, counted at 2.0 snoise evaluations (an estimate, printed beside the count).

Consumers: the gamecult.org ground shader (through the GLSL lowering) and Aetheria's nebula raymarch at its CultMath pin bump.

Affine forms

math.Affine.cs carries reduced affine forms with one shared symbol: a float3(x0, a, e) is the set x0 + a eps + e delta with eps and delta in [-1, 1] and e >= 0 (Messine's AF1, as Gamito and Maddock use it for gradient noise, in one register as an interval is a float2). eps is shared by every form over one region (for a screen-tile slice, the depth parameter, z = z_m + h eps), so af_sub(x, x) has a = 0; delta is private to the form, and every term that cannot be tracked is condensed into e. A slice is a float4(centre, radius) with a float3 axis. No op takes a compound type, a reference parameter or a swizzle write, so each mirrors into the HLSL common subset.

The ops: af_point, af_symbol, af_from_iv, af_range (the bridge back to iv_*), af_add, af_sub, af_neg, af_scale, af_add_iv, af_mul (the product |a_x a_y| and the cross-error terms condensed into e, so there is no af_sqr), af_snoise and af_fbm, and the slice pair af_frustum_axis and af_frustum_ball.

Rounding. Every op returns a form whose set at each eps contains the float32 value of its function at every point of its operands' sets at that eps. Each op folds its own float32 rounding into e as S 2^-20, S the sum of the magnitudes its roundings are relative to (named in the op's comment): at most 8u S is needed with u = 2^-24, and the factor of two over that is the allowance for a device whose + and * are faithful rather than correctly rounded. AffineTests.EveryOpEnclosesItsPoints evaluates the form's set in double and the function in float32 with no tolerance. Products and af_from_iv also add 2^-126, the most an underflow to a subnormal can lose (TinyOperandsEncloseTheirPoints). af_from_iv halves its endpoints before it sums and differences them, and af_range scales its terms before it sums them, so endpoints near the float maximum do not overflow in the middle of the op; a form that overflows anyway gives af_range infinite endpoints, never NaN (FromIntervalAndRangeHoldAtTheExtremes). af_range finds a NaN endpoint by its bit pattern, because lo == lo is a compare dxc marks fast and deletes under its default flags. AbsorptionsArePinned and SnoiseFormCarriesItsDocumentedWidth pin each absorption and af_snoise's float32 allowance both ways: the enclosure tests alone cannot see them on a CPU, where float32 is exact.

af_snoise(centre, axis, radius) encloses snoise over centre + axis eps + w, |w| <= radius, from one snoise_grad. It returns the narrower of the Lipschitz form (n, 0, L R) and the centred form (n, g . axis, |g| radius + M R^2 / 2) (R = |axis| + radius, L = SNOISE_LIPSCHITZ, M = SNOISE_HESSIAN), so its total width is never more than the Lipschitz form's. The centred form is Taylor's theorem, which is why it beats the ball when the radius is small and the axis carries the variation. af_fbm compounds octaves as iv_fbm_ball does. The float32 evaluation allowance is derived in af_snoise's comment.

SNOISE_HESSIAN = 55.642143 is 1.10 times the refined maximum of the spectral norm of the Hessian of snoise(float3), measured by NoiseBoundTests.MeasureHessian over 1e6 seeded points (the analytic snoise_hessian, below) and refined by ascent from the 1e4 largest: refined maximum 50.5838. The central differences of snoise_grad that an earlier pin used read 50.955 at h = 1e-3, their roundoff. HessianConstantPinsSampledCurvature fails on a 1% move. snoise_hessian(float3) (math.Span.cs) is the analytic Hessian of snoise: a symmetric float3x3 on the same lattice, hash and tie order as snoise_grad, differentiating each corner's gradient -8 m^3 s x + m^4 p once more. Each off-diagonal entry is computed once and placed twice, so the matrix equals its transpose bit for bit, and a clamped corner (m = 0) contributes nothing. Built from nine scalars, the matrix reads as columns in GLSL (mat3) and rows in HLSL, which is the same matrix because it is symmetric. SNOISE_THIRD = 535.8009 (math.Span.cs) is the next order: 1.10 times the maximum, over 1e6 seeded points, of the operator norm of the third-derivative tensor of snoise, refined from the 1e4 largest; measured 487.0917 on the tied kernel. The tensor is closed form, in double: snoise_hessian's per-corner entry differentiated once more, on the kernel's lattice, tie order and hash, and the test ThirdTensorMatchesDifferencesOfTheHessian holds it against Richardson-extrapolated differences of the double Hessian. The operator norm is a local ascent over unit vectors (a dense sweep of 20,000 directions at the witness finds 99.9987% of it) and the outer maximum a sampled search refined by ascent, so the constant is empirical with a 10% margin, not a proof; an earlier pin from central differences of the float Hessian read 488.1047, its roundoff. MeasureThird takes the measurement and ThirdConstantPinsSampled pins it. integral_clamp_linear(a, b, lo, hi, t0, t1) is the closed-form integral over [t0, t1] of clamp(a + b t, lo, hi). All three have HLSL and GLSL lowerings and glsl-parity.json fixture families (snoise_hessian, integral_clamp_linear; the af_snoise and af_fbm families are regenerated for the re-pin). SNOISE2_LIPSCHITZ = 8.117002 is the same for the gradient of snoise(float2) (MeasureLipschitz2: sampled 7.2575, refined 7.3791, witness (-67.12454, -129.53296), |snoise2| at most 0.99964), and it backs iv_snoise_ball(float2, float). Both are empirical with a margin, not proofs. A random ball cannot see a curvature constant 20% low, so SNOISE_HESSIAN is defended by HessianConstantPinsSampledCurvature and SnoiseFormEnclosesTheCurvatureWitness (af_snoise at the point where the curvature is realised, along 4,000 directions: a constant of 40 misses there). The margin over the function itself: a scan of the whole 289^3 hash period found a realised maximum of 50.58, and bounding each corner's gradient independently over every hash configuration gives 52.22, so 55.64 sits 7% above the configuration bound. A change to the kernels must re-run the measurements.

Span stages

span_fbm and span_ridged (math.Span.cs; cultmath_span_fbm and cultmath_span_ridged in HLSL over the same struct types, cultmath_span_stage behind both, and GLSL derived) are the octave stages of a warp chain, span_fbm(path, stage, octaves, lacunarity, gain), each returning one SpanNoise { float4 field; float curvature; float fold; }. The inputs are two structs with named public fields and no positional constructor, so the three floats of the span and the six fields of the transform cannot be swapped silently. SpanPath { midpoint, velocity, speed, acceleration, halfLength } is the path q(t), |t| <= halfLength, through midpoint with velocity at t = 0, |q'| <= speed and |q''| <= acceleration along it. SpanStage { frequency, stretch, axis, shift, offset, octaveStep } makes octave o sample snoise_grad at x_o(q) = f_o (M q + shift) + offset + o octaveStep, with f_o = lacunarity^o, amplitude gain^o, and M = frequency (I + (1/stretch - 1) a a^T), a = normalize(axis); a zero axis or stretch 1 means no stretch, and stretch must be positive and finite (unguarded, like lacunarity). default(SpanStage) has frequency 0; SpanStage.Identity is the stage whose field equals fbm_grad and ridged_grad bit for bit. octaves, lacunarity and gain stay positional: they are fbm_grad's own trailing triple, and octaves is the one int among the scalars. field is the octave sum and its gradient in q (f_o M grad n_o). span_ridged sums gain^o (1 - |n_o|), ridged_grad's convention. The Span Lab's stage is 1 - 2|n| per octave: with G the sum of gain^o over the octaves (signed, so the alternating sum for a negative gain), its value is `2 value

• G, and its gradient, curvature and fold are twice span_ridged`'s.

The shaders have no SpanStage.Identity: a shader caller builds every stage field itself (in GLSL, SpanStage(1.0, 1.0, vec3(0.0), vec3(0.0), vec3(0.0), vec3(0.0)) is the identity). A constant would need a second transformation rule (C# has no const structs, GLSL wants a constructor) and a mirror exemption, for one value; the mirror test proves instead that the shader text at a C#-built identity returns cultmath_fbm_grad and cultmath_ridged_grad's field bit for bit.

Bounds come from the three pins only. With |M| = |frequency| max(1, 1/stretch) and nu = f_o |M| speed the noise-space speed, the Hessian of the noise on the span has operator norm at most Hl = min(SNOISE_HESSIAN, ||snoise_hessian(x_mid)||_F + e_h + SNOISE_THIRD nu halfLength), so curvature = sum gain^o f_o^2 |M|^2 Hl bounds the operator norm of the field's Hessian in q at every point of the span (the pin takes over once the third-derivative term exceeds it, and at halfLength 0 the local tier alone remains). For span_ridged, an octave leaves the branch it is on at the midpoint (1 - s n, s the midpoint's sign of n) only where n changes sign, and there |n| <= V - |n_mid|, with `V = (|grad n_mid . w| + e_g |w|) halfLength + (Hl nu |w|

• Gl f_o |M| acceleration) halfLength^2 / 2 + 2 e_n, w = f_o M velocityandGl = min(SNOISE_LIPSCHITZ, |grad n_mid| + e_g + Hl nu halfLength); fold = sum 2 gain^o max(V - |n_mid|, 0), exactly 0 for span_fbm` and for an octave that cannot reach zero.

The float32 snoise, snoise_grad and snoise_hessian are not the exact functions, so the quantities read at the midpoint carry allowances, by the derivation of af_snoise's comment. S_o = |f_o| (|M q|_1 + |shift|_1) + |offset|_1 + |o octaveStep|_1 + sqrt(3) nu halfLength is the sum of the magnitudes the octave coordinate's roundings are relative to, and e_n = 2^-14 + L 2^-20 S_o (value), e_g = 2^-12 + H 2^-20 S_o (gradient length) and e_h = 2^-10 + T 2^-20 S_o (Frobenius norm of the Hessian), with L, H, T the three pins; they add no pin. Measured against a double kernel with the float hash (6000 points per band, |x| from 2^-2 to 2^23), two samples observed the error's maxima as fractions of e_n, e_g and e_h: 0.057, 0.046 and 0.075 in the cut's own sample (seed not recorded; the 0.075 is against T 2^-20 |x|_1 alone, at the smallest band, where the absolute Hessian error is 2.3e-5 and the absolute term carries it), and 0.064, 0.0485 and 0.057 in an audit's (seed 0x501). They are samples, not bounds: a new seed moves them. They say the allowances enclose the error with a margin of about 15. Errors grow linearly in |x|: the value's is 2e-4 at 2^8, 1.5e-2 at 2^14 and 0.94 at 2^20. Rounding in the Frobenius-to-operator-norm step needs no allowance: the Frobenius norm exceeds the operator norm.

Domain. No coordinate range is checked, clamped or refused. The allowances grow with S_o, so curvature saturates at SNOISE_HESSIAN (true everywhere) by S_o of about 1.1e5, and fold turns positive for every octave whose V + 2 e_n reaches |n_mid|: the bounds stay sound at every magnitude and stop being informative where float32 noise is (see "Precision domain" above). SpanStageTests encloses the exact (double) Hessian norm and both the float and the exact branch departure at 64 points of seeded spans in bands of S_o from under 1e2 to over 1e6, and in a fixed span where the local tier with its third-derivative term halved fails.

The slice. af_frustum_ball(centreSlope, z0, z1, footprintPerDepth, warpVariation) returns the centre (m_c z_m, z_m) and a radius z1 footprintPerDepth + warpVariation, with af_frustum_axis the axis (m_c, 1) h: every ray point at depth z is centre + axis eps(z) + w. Its derivation and the clamp residual are in the comment. The caller rotates, translates and scales the slice into the noise's frame, and owns the rounding of that.

The centred warp. A flow-warped sample reads snoise at p + flow(p) shift. The caller does not enlarge the ball by the whole displacement |flow| shift; it moves the centre by the flow at the centre, c + flow(c) shift, and widens the radius by the flow's variation across the ball, r + |shift| rho with rho = FlowAmplitude (SNOISE2_LIPSCHITZ FlowFrequency r + a) sqrt 2 (the clamp to the unit disc is nonexpansive), where a = 2^-14 + L2 2^-20 (|q|_1 + r FlowFrequency) is the float32 allowance of the flow's two snoise(float2) reads (the point q plus its offset rounds at the magnitude of q, and a float32 flow at two points of a ball differs by more than the exact function can). CultMath owns neither the flow nor the envelope: the warp's centre and variation are the caller's, and a consumer adds them to the slice's warpVariation or to the form's radius. With two phases that cross-fade, each phase takes its own centre; one ball serves when the phases' centres coincide (warp 0). NoiseBoundTests.CentredWarpStaysEnclosed checks 2,000 balls, radius 1e-3 to 500, and fails when rho is dropped or halved; AffineWarpStaysEnclosed checks 3,000 tile slices of FogField.Slice, which returns the final af_snoise arguments (each phase's moved centre and grown radius, the detail's unwarped set, the shared axis), so the affine bound wires nothing itself: the reach over which the flow is varied is |axis| + radius, each phase carries its own growth, and the gate reads each argument's reach. It fails when the reach is the ball's radius or the larger of the two, a phase's growth is dropped or taken from the other phase, the axis or the detail radius is scaled, or rho is halved. AffineCompositionTightensItsRanges sees the shared-symbol composition itself: the bound is exactly the intersection of the affine arithmetic's piece and the interval composition over the forms' ranges, and the affine piece is the narrower of the two in at least 35% of the open slices (the piece alone is wider in total).

The deep well is the representative Aetheria case: the camera inside a sun's bowl, looking across. DeepWellIsTenBandsDeepAndWide computes the band G from FogField's own floor offset, amplitude and noise range (60) and requires the depth (633 at the centre against the rim) and the half-depth radius to be at least 10 G, every drawn camera sitting inside the well below the rim; a well of depth 150 fails. The shallow wells stay as the printed shallow-well, band-dominated row (c-band).

The test field's affine bound takes depth as af_symbol(z_m, h), builds s affine in it with the height interval, the footprint and the slope-depth product in e through af_add_iv, takes the noise through af_frustum_* and af_snoise per phase and for the detail octave, fades by af_from_iv and af_mul, reads the range with af_range, and intersects with the centred interval's bound (both sound). FogField.Bound makes no direct snoise call. The cull-map render (DeepWellCullMap, explicit) writes a PNG when CULTMATH_WRITE_CULLMAP names a directory: one deep-well camera at warp D, 960 x 540 in 8 x 8 tiles, with panels for dense cells per ray under the centred interval, under the affine bound and under the oracle mask (one logarithmic scale) and the transmittance. Its run: 4.23, 3.60 and 2.02 cells per ray on average, 44 the most, mean transmittance 0.278.

Expressions

CultMath.Expr holds a scalar SSA expression graph as a persisted value, a builder for it, and a C# evaluator.

• Graph (ExprGraph): a sealed immutable class. It has no name of its own; the consumer record that holds it names it. Nodes are in topological order, an operand names a lower node, and an edit builds a new graph. A persisted graph carries a format version (1); one naming an unknown op, a newer format or a forward operand is refused whole at decode. ExprGraph.Create is the one validator, run by the builder and by decode.
• Node: its index in the topological order. The builder hash-conses identical (op, operands, constant bits) nodes and drops nodes no output reaches, so one expression always builds the same bytes; any rebuild may renumber nodes.
• Port: graph inputs vary per point, params are uniform per batch and carry a default, outputs name a node. Names are [A-Za-z_][A-Za-z0-9_]*, unique per kind. An op's operands are positional; a multi-result op (cellular3: nearest.xyzw, edge.xyzw, id) is read only through Pick.
• Op vocabulary: one table (ExprOps), a stable string per op, append-only. Vector arguments are decomposed operands: snoise2, snoise3, cellular3. An op calls its CultMath function; a function's value change is a Breaking release note.
• Evaluation: ExprEvaluator.Evaluate runs the nodes in order through the table, one IEEE single per op, so a result equals the direct composition of the same math calls bit for bit. Params bind by name (ExprBindings): a missing name takes its default and an unknown name is refused.
• Stencils are substitution at build time (ExprBuilder.Substitute, Fidget's remap_xyz): N substitutions of the position inputs combined by ops give an ordinary flat graph. There is no loop construct.

The MessagePack shape belongs to GameCult.Caching.MessagePack (CultMathExprResolver). Interval, affine and gradient evaluation over the graph, bytecode, the HLSL interpreter and pruning are not built.

Rules

• Keep type names HLSL-shaped: float2, float3, float4, float3x3, and math.
• Keep Rust as the owner of native hot-path kernels and parity fixtures.
• Keep C#, shader, and scripting surfaces as runtime wrappers, not competing math authorities.
• Keep shaders/CultMath.hlsl as the canonical shader mirror for shared CultMath functions HLSL does not already provide (HLSL intrinsics such as mul, any, and log need no mirror). Project-local shader helpers should move here once more than one organ needs them or once CPU/GPU parity matters.
• Implement public shader semantics directly; do not copy Unity.Mathematics.
• Prefer plain mutable value types with public component fields, as HLSL does.
• Prefer component-wise overloads over clever generic machinery until the API earns that complexity.
• Keep the surface boring and exact. Fill what a consumer uses, guided by HLSL semantics; noise primitives, matrices, quaternions, and packing helpers need an owner before they enter.
• Keep the core assembly engine-free. Unity conversions live in the Unity package's CultMath.UnityBridge assembly.

Non-Goals

• Compiling C# to shaders.
• Generating HLSL from Rust or Rust from HLSL.
• Replacing System.Numerics for general .NET work.
• Recreating every Unity.Mathematics type because a spreadsheet somewhere says “coverage.” That way lies decorative obesity.